Vector Space Properties
Some important properties of vector space are:
- Closure under Addition: Sum of any two vectors in the vector space is also a vector in the vector space.
- Closure under Scalar Multiplication: Multiplying any vector in the vector space by a scalar yields another vector in the vector space.
- Associativity of Addition: Vector addition is associative, meaning (u + v)+ w = u + (v + w) for all vectors u, v, and w in the vector space.
- Commutativity of Addition: Vector addition is commutative, meaning u + v = v + u for all vectors u and v in the vector space.
- Existence of Additive Identity: There exists a vector, denoted by 0 or 0, called the zero vector, such that u + 0 = u for all vectors u in the vector space.
- Existence of Additive Inverse: For every vector u in the vector space, there exists a vector -u such that u + (-u) = 0.
- Distributive Properties: Scalar multiplication distributes over vector addition, meaning α(u + v) = αu + αv and (α+β)u = αu + βu for all scalars α and β, and vectors u and v in the vector space.
- Multiplicative Identity: Scalar 1 acts as the multiplicative identity, meaning 1⋅u = u for all vectors u in the vector space.
Vector Space- Definition, Axioms, Properties and Examples
A vector space is a group of objects called vectors, added collectively and multiplied by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces.
In this article, we have covered Vector Space Definition, Axions, Properties and others in detail.
Table of Content
- What is Vector Space?
- Vector Space Axioms
- Vector Space Examples
- Dimension of a Vector Space
- Vector Addition and Scalar Multiplication
- Vector Space Properties
- Subspaces
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